I am a double major in finance and statistics from UT austin and nobody is even interested in interviewing me for an entry level position in accounting/analysis because I have no work references. My previous jobs have been temp contract jobs where I have been the top performer on my team (unrelated to my field). However my supervisors say for legal reasons they are not allowed to be a work reference. I have several other accomplishments and even picked up coding skills in java/c+/sql and nobody cares about that. Even for a $13/hr entry level position they only care about work references.
Why can't you use previous professors as reference? When I got my first job, I used some professors that I had done work for along with other people that I done work for. I also recommend building up a portfolio of your skills. A good idea going forward is to volunteer your skills to a church or a not for profit organization. That way you can build up references.
It's unfortunate that schools/universities/colleges aren't teaching students how to get a job. I've known people just sit there, do their homework, attend classes, then graduate without a job. They continue to not have any jobs even after years. So your case is not uncommon. A good idea to become active in some way. Ask professors for side works, projects, or anything. Do minor jobs for someone who you can reference. Also intern work is a huge plus.
References are just a way of making sure that you can play nice with others in a work setting. Generally, references from work are preferred if not explicitly required, but you can also use your colleagues and if your work direct supervisor is unable to or unwilling. You could also use a former professor or classmate that you worked on a project with.
Not all jobs will require references though. If you keep applying to jobs, you're sure to find some.
So, the problem isn't really that you have no references, it's that 2 years out from graduation you have no work references.
Rightly or wrongly there will be an assumption that you have had trouble getting into being a worker in the business world.
I would look to finding someone (a relative or better a professor or family friend in a position in business) who can give you a personal reference. If asked I'd offer that (but be honest and explain why you have trouble and who the referee is). Make sure it is someone who can vouch for you working (your Mum won't cut it).
Failing that, get involved in volunteering, and see if you can get to where a supervisor could provide you with one, that'll take time, but if you have no other way it's better than nothing.
Also look at intern roles, although you'll have competition from fresh graduates and a similar problem (why are you looking 2 years down).